Page 137 - Phase Space Optics Fundamentals and Applications

118 Chapter Four

function is converted to a two-dimensional object by the use of cylin-
drical lenses to allow the construction of a multichannel processor that
optically implements the calculations of the RWD. The setup consists
of three phase masks separated by fixed distances in free space. The
masks consist of many strips, each one representing a different chan-
nel that performs an FrFT with a different order over the input signal.
Each strip is a Fresnel zone plate with a different focal length that is
selected for obtaining the different fractional orderp. Thus, the main
shortcoming of the RWD chart produced by this setup is that it has a
limited number of projection angles (or fractional orders). Besides the
very poor angular resolution, the experimental results obtained in the
original paper are actually very far from the theoretical predictions.
A truly continuous display, i.e., a complete RWD setup, was pro-
6
posed by Granieri et al. This approach is based on the relationship
between the FrFT and Fresnel diffraction, 7,8 which establishes that
every Fresnel diffraction pattern of an input object is univocally
related to a scaled version of a certain FrFT of the same input. There-
fore, if the input function f (x) is registered in a transparency with
amplitude transmittance t(x/s), with s being the construction scale
parameter, then the FrFT of the input can be optically obtained by
free-space propagation of a spherical wavefront impinging on it.
Actually, the Fresnel diffraction field U(x, R p ) obtained at distance
R p from the input, which is illuminated with a spherical wavefront
of radius z and wavelength , is related to the FrFT of order p of the
input function F {t (x) , } as follows: 9
p
2
i x z(1 − M p ) − R p p x
U(x, R p ) = exp F t ,x (4.41)
zR p M 2 p M p
where M p is the magnification of the optical FrFT. For each fractional
order, the values of M p and R p are related to the system parameters
s, , and z through
2 −1
s tan( p /2)
R p = 2 −1 (4.42)
1 + s (z ) tan( p /2)
1 + tan( p /2) tan( p /4)
M p = (4.43)
2
1 + s (z ) −1 tan( p /2)
These last equations allow us to recognize that by illumination of an
input transparency with a spherical wavefront converging to an axial
point S, all the FrFTs in the range [0, 1] can be obtained simultaneously,
apart from a quadratic-phase factor and a scale factor. The FrFTs are
axially distributed between the input transparency plane ( p = 0) and
the virtual source (S) plane ( p = 1) in which the optical FT of the input
is obtained. For one-dimensional input signals, instead of a spherical